论文标题
分数laplacian的局部能量估计
Local energy estimates for the fractional Laplacian
论文作者
论文摘要
（0,1）$的订单$ s \的整体分数拉普拉斯是一个非本地运营商。众所周知，涉及此类操作员的Dirichlet问题的解决方案表现出代数边界的奇异性，无论域的规律性如何。反过来，这会恶化解决方案的全球规律性，结果是数值解决方案的全球收敛速率。对于有限元离散化，我们在$ h^s $ seminorm中得出局部误差估计，并通过仅假设网格为形状定型，在域内显示了最佳的收敛速率。这些估计值量化了降低近似误差集中在域边界附近的事实。我们用几个数字示例来说明我们的理论结果。
The integral fractional Laplacian of order $s \in (0,1)$ is a nonlocal operator. It is known that solutions to the Dirichlet problem involving such an operator exhibit an algebraic boundary singularity regardless of the domain regularity. This, in turn, deteriorates the global regularity of solutions and as a result the global convergence rate of the numerical solutions. For finite element discretizations, we derive local error estimates in the $H^s$-seminorm and show optimal convergence rates in the interior of the domain by only assuming meshes to be shape-regular. These estimates quantify the fact that the reduced approximation error is concentrated near the boundary of the domain. We illustrate our theoretical results with several numerical examples.
